Peirce's Graphs (en)
Peirce's Graphs (en)
Über mich (ge)
The focus of my research is formal languages which provide a formal means, based on mathematical graphs, to visually represent and process information and to carry out mathematical logic.
My PhD-thesis `The Logic System of Concept Graphs with Negations (And its Relationship to Predicate Logic)', which I wrote at Darmstadt Technical University (Germany) under the supervision of Prof. Wille, discusses mathematical structures based on John Sowa's conceptual graphs and Rudolf Wille's formal concept analysis. Conceptual graphs in turn are based on Peirce's existential graphs and the semantic networks of artificial intelligence. Conceptual graphs form a diagrammatic logic system, which is used in, but not limited to, the fields of natural language processing and knowledge representation. Their purpose is `to express meaning in a form that is logically precise, humanly readable, and computationally tractable' (Sowa). Wille's formal concept analysis (FCA) is a mathematical theory which is mainly used for the conceptual evaluation of data and information. With the use of several FCA-tools, it has been successfully used in some hundred projects. In my PhD-thesis, a fragment of conceptual graphs with equivalence to first order predicate logic is mathematically elaborated. Thus it bridges the gap between conceptual graphs, known from computer science, and mathematical logic. The thesis has been published by Springer in the series `Lecture Notes in Artificial Intelligence', vol. LNCS 2892, in November 2003.
Another diagrammatic system I have thoroughly scrutinised is Peirce's existential graphs. These graphs are a diagrammatic version of first order logic, similar to concept graphs with negations. Peirce (1839-1914) developed this system starting at the end of the 19th century. Although there is some amount of secondary literature, a comprehensive mathematical elaboration of existential graphs has not yet been provided. In a nearly finished treatise which is planned to be my habilitation thesis, the system of existential graphs is formally elaborated.
Compared to my PhD-thesis, the goal of my habilitation thesis is significantly broader. First of all, as Peirce's description of existential graphs by no means suit the needs of contemporary formal elaborations of logic, a careful study of his original writings is needed. Second, among philosophers, Peirce is better known for his pragmatism and his comprehensive semiotics. Nonetheless, he called his system of existential graphs his `chef d'oeuvre'. For this reason, I investigated the place of Peirce's graphs in his overall philosophy. Most importantly, my thesis aims at a general methodology for a mathematical foundation of diagrammatic logic. Besides his system of existential graphs, Peirce contributed to the development of symbolic logic to a large extent (for example, independently from Frege, Peirce and one of his students invented a notion for quantification). Driven by his vast theory of signs and his understanding of human reasoning, Peirce dropped this symbolic approach and developed his diagrammatic system of existential graphs instead. Based on Peirce's works, important differences between symbolic and diagrammatic approaches to formal logic are investigated in my treatise, and a methodology for a formal theory of logic with diagrams is developed. In this framework, the mathematical elaboration of existential graphs can be considered to be a case study for the more general approach.
My core research, as described above, is extended in the following ways. First of all, I am interested in further elaborating the formal foundations of conceptual and existential graphs. An example is the addition of so-called `nestings', which allow meta-statements within the language of concept graphs, to the syntax and semantics of conceptual graphs. Moreover, I worked out some interesting properties of proofs within the system of existential graphs, and I extended the syntax of existential graphs to capture constants, functions, and free variables. Second, I am interested in other formal diagrammatic reasoning systems as well. Examples are spider diagrams and constraint diagrams. Finally, and most importantly, I am interested in applications of conceptual and existential graphs in other fields of knowledge representation and processing. I am particularly interested in formal languages originating from the semantic web. I have investigated RDF in terms of a formal diagrammatic reasoning system, and I am currently working on elaborating different description logics in a diagrammatic manner (the description logic ALC is finished). Besides that, I am working with a colleague on an approach to diagrammatically explore and extend XML-documents w.r.t. a given grammar. Another application that I am exploring is using conceptual graphs as a diagrammatic query language for relational databases.
There exists a variety of languages which allow diagrammatic representation and processing of information. Well-known examples in information technology are: UML (which is not intended to process information), RDF (Resource Description Framework), being one of the underlying languages of the semantic web, topic maps, and semantic networks, which are used in artificial intelligence or knowledge engineering. Although these languages are already used --including inference processes-- in many applications, a comprehensive mathematical investigation or foundation of them is often missing. On the other hand, there exists a variety of languages which are thoroughly formally elaborated, usually based on mathematical logic, but which rely on symbolic notations (e.g., description logics). As a result we have a gap between (a) diagrammatic representation languages, which are easier to comprehend but which lack precisely defined semantics and reasoning facilities, and (b) formal logics, which provide mathematically solid syntax, semantics and reasoning facilities, but which due to their symbolic notation are hard for untrained users to comprehend. The goal of my research can now be sketched as follows:
to develop a human-centered, mathematically precise framework for the diagrammatic representation and processing of information and knowledge.
Particularly, my research is not intended to compete with the existing symbolic approaches to formal languages. Instead, it should be understood to complement them.
Letzte Änderung / last change: 2023-04-07